Model-independent determination of the two-photon exchange contribution to hyperfine splitting in muonic hydrogen

Abstract

We obtain a model-independent prediction for the two-photon exchange contribution to the hyperfine splitting in muonic hydrogen. We use the relation of the Wilson coefficients of the spin-dependent dimension-six four-fermion operator of NRQED applied to the electron-proton and to the muon-proton sectors. Their difference can be reliably computed using chiral perturbation theory, whereas the Wilson coefficient of the electronproton sector can be determined from the hyperfine splitting in hydrogen. This allows us to give a precise model-independent determination of the Wilson coefficient for the muonproton sector, and consequently of the two-photon exchange contribution to the hyperfine splitting in muonic hydrogen, which reads δĒ TPE pµ,HF (nS) = − 1 n 3 1.161(20) meV. Together with the associated QED analysis, we obtain a prediction for the hyperfine splitting in muonic hydrogen that reads E th pµ,HF (1S) = 182.623(27) meV and E th pµ,HF (2S) = 22.8123(33) meV. The error is dominated by the two-photon exchange contribution.

The next generation of laser spectroscopy experiments using light muonic atoms

“…Eq. (9) was obtained from the values given in[59] and assuming a proton Zeemach radiuis of r 1 H Z = 1.045(4) fm[82]. Hereby we conservatively assumed that the uncertainty of the polarizability contribution (included in the first term in Eqs (8).…”

“…Hereby we conservatively assumed that the uncertainty of the polarizability contribution (included in the first term in Eqs (8). and(9)) is equal to the uncertainty of the total TPE of[59].…”

Laser spectroscopy of light muonic atoms and the nuclear charge radii

“…The goal is to measure this transition with 1-2 ppm precision from which the 2PE contribution can be obtained with 10 −4 relative accuracy. The extracted 2PE contribution can be then compared to predictions from chiral perturbation theory (chPT) or from data-driven (proton structure functions and form factors) dispersion relations [11,61,62].…”

Laser excitation of the 1s-hyperfine transition in muonic hydrogen

“…From the measurement of the HFS, precise information about the magnetic structure of the proton can be extracted [13][14][15][16][17][18][19][20][21][22][23]. Specifically, by comparing the measured HFS transition frequency with the corresponding theoretical prediction based on bound-state QED calculations [5,13,19,20], the two-photon-exchange contribution can be extracted with about 2 × 10 −4 relative accuracy.…”

“…From the measurement of the HFS, precise information about the magnetic structure of the proton can be extracted [13][14][15][16][17][18][19][20][21][22][23]. Specifically, by comparing the measured HFS transition frequency with the corresponding theoretical prediction based on bound-state QED calculations [5,13,19,20], the two-photon-exchange contribution can be extracted with about 2 × 10 −4 relative accuracy. Because the two-photon-exchange contribution can be expressed as the sum of a finite-size (static, elastic) part proportional to the Zemach radius (R Z ) and a polarizability part (dynamic, virtual excitation), its determination can be used to extract separately the two parts: the Zemach radius when the polarizability contribution is assumed from theory [13,15,16,18,[21][22][23][24][25], and the polarizability contribution when taking R Z from electron-proton scattering or hydrogen [19,[26][27][28].…”